It's the first stop on his "World Tour," as he labeled it in an interview with USA Today Sports last week. He said he'll "be going Monday to Friday, working out for different teams."
Smith is scheduled to arrive in Kansas City later Monday. He will spend extensive time with the Chiefs' staff Tuesday.
The Chiefs hold the No. 1 overall pick and new coach Andy Reid said his team has not closed the door on picking Smith in that spot.
"I hope Kansas City is legitimately interested. I hope I go No. 1, but, hey, we'll see," Smith told USA Today on Friday in a telephone interview. "The Chiefs are bringing me in, so I'm going in there thinking that they're interested [in me], because I am in them."
Smith threw for 4,205 yards and led the nation with 42 touchdown passes for the Mountaineers this past season.
He impressed at West Virginia's pro day March 14, completing 60 of 64 throws in a workout attended by all but three NFL teams.
Smith told USA Today that he will meet again with the Bills this week. It's unknown what level of interest Buffalo still has in selecting Smith, however, after agreeing to a two-year, $13 million contract with quarterback Kevin Kolb on Saturday.
Smith also told USA Today that he met with New York Jets offensive coordinator Marty Mornhinweg and executive Terry Bradway before his pro day, but nothing "extensive" was discussed.
Jacksonville Jaguars coach Gus Bradley and Eagles owner Jeffrey Lurie and coach Chip Kelly are among the NFL decision-makers to give positive reviews about the quarterback after meeting with him.
"All of the teams have been really impressed with me, they've all said good things," Smith told USA Today. "I'm not going to put my hope or my eggs in one team's basket. All of them have shown a high level of interest. It's not a case where I know of one team being more interested. We'll just have to see on draft night."
Information from ESPN Senior NFL Insider Chris Mortensen, ESPN NFL Insider Adam Schefter and The Associated Press was used in this report.